The local density approximation is used to study the ground state superfluid properties of harmonically trapped p-wave Fermi gases as a function of fermion-fermion attraction strength. While the density distribution is bimodal on the weakly attracting BCS side, it becomes unimodal with increasing attraction and saturates towards the BEC side. This non-monotonic evolution is related to the topological gapless to gapped phase transition, and may be observed via radio-frequency spectroscopy since quasi-particle transfer current requires a finite threshold only on the BEC side.
We use the Bogoliubov-de Gennes formalism to analyze the effects of rotation on the ground state phases of harmonically trapped Fermi gases, under the assumption that quantized vortices are not excited. We find that the rotation breaks Cooper pairs that are located near the trap edge, and that this leads to a phase separation between the nonrotating superfluid (fully paired) atoms located around the trap center and the rigidly rotating normal (nonpaired) atoms located towards the trap edge, with a coexistence (partially paired) region in between. Furthermore, we show that the superfluid phase …
We investigate trapped resonant fermions with unequal populations within the local density approximation above the superfluid transition temperature. By tuning the attractive interaction between fermions via Feshbach resonance, the system evolves from weakly interacting fermi gas to strongly interacting fermi gas, and finally becomes bose-fermi mixture. The density profiles of fermions are examined and compared with experiments. We also point out the simple relationships between the local density, the axial density, and the gas pressure within the local density approximation.
The moment of inertia Q of a trapped superfluid gas of atomic Fermions (6Li) is calculated as a function of the temperature. At zero temperature the moment of inertia takes on the irrotational flow value. Only for T very close to Tc rigid rotation is attained. It is proposed that future measurements of the rotational energy will unambiguously reveal whether the system is in a superfluid state or not.
In a very interesting recent Letter\cite{machida}, the authors suggested the possibility of realizing the spatially modulated, or Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid state in trapped atomic fermion systems. The authors this Letter used a 1D mean field solution as guidance to estimate the parameter range for the existence of the FFLO phase, and also discussed the possibility of its detection by imaging the atomic density of the system. In this comment I wish to make two points. (i) In 1D there exists an exact solution based on bosonization, which fully takes into account the impo…
From an extensive calculation of static properties of a trapped Fermi superfluid at zero temperature using a density-functional formulation, we demonstrate a universal behavior of its observables, such as energy, chemical potential, radius etc., over the crossover from the BCS limit to unitarity leading to scaling over many orders of magnitude in fermion number. This scaling allows to predict the static properties of the system, with a large number ($\sim 10^5$) of fermions, over the crossover with an error of 1-2%, from the knowledge of those for a small number ($\sim 10$) of fermions.
We consider the BCS-BEC crossover for a system of trapped Fermi atoms at finite temperature, both below and above the superfluid critical temperature, by including fluctuations beyond mean field. We determine the superfluid critical temperature and the pair-breaking temperature as functions of the attractive interaction between Fermi atoms, from the weak- to the strong-coupling limit (where bosonic molecules form as bound-fermion pairs). Density profiles in the trap are also obtained for all temperatures and couplings.
We consider a superfluid of trapped fermionic atoms and study the single vortex solution in the Ginzburg-Landau regime. We define simple analytical estimates for the main characteristics of the system, such as the vortex core size, temperature regimes for the existence of a vortex, and the effects of rotation and interactions with normal fermions. The parameter dependence of the vortex core size (healing length) is found to be essentially different from that of the healing length in metallic superconductors or in trapped atomic BEC in the Thomas-Fermi limit. This is an indication of the import…
We study the properties of a trapped interacting three component Fermi gas. We assume that one of the components can have a different mass from the other two. We calculate the different phases of the three component mixture and find a rich variety of different phases corresponding to different pairing channels, and simple ways of tuning the system from one phase to another. In particular, we predict co-existence of several different superfluids in the trap, forming a shell structure, and phase transitions from this mixture of superfluids to a single superfluid when the system parameters or tem…
We examine the effect of the trapping potential on the critical temperature, $T_C$, for the BCS transition to a superfluid state in trapped atomic gases of fermions. $T_C$ for an arbitrary power law trap is calculated in the Thomas-Fermi approximation. For anharmonic traps, $T_C$ can be increased by several orders of magnitude in comparison to a harmonic trap. Our theoretical results indicate that, in practice, one could manipulate the critical temperature for the BCS phase transition by shaping the traps confining the atomic Fermi gases.
It is found theoretically based on the Ginzburg-Landau framework that p-wave superfluids of neutral atom gases in three dimension harmonic traps exhibit spontaneous mass current at rest, whose direction depends on trap geometry. Under rotation various types of the order parameter textures are stabilized, including Mermin-Ho and Anderson-Toulouse-Chechetkin vortices. In a cigar shape trap spontaneous current flows longitudial to the rotation axis and thus perpendicular to the ordinary rotational current. These features, spontaneous mass current at rest and texture formation, can be used as diag…
We present phase diagrams for population imbalanced, trapped Fermi superfluids near unitarity. In addition to providing quantitative values for the superfluid transition temperature, the pairing onset temperature and the transition line (separating the Sarma and phase separation regimes), we study experimental signatures of these transitions based on density profiles and density differences at the center. Predictions on the BCS side of resonance show unexpected behavior, which should be searched for experimentally.
Two issues are treated in this work: (i) the generic fact that if a fermionic superfluid in the BCS regime overflows from a narrow container into a much wider one, pairing is much suppressed at the overflow point. Physical examples where this feature may play an important role are discussed (ii) A Thomas-Fermi (TF) approach to inhomogeneous superfluid Fermi-systems is presented and shown that it works well in cases where the Local Density Approximation (LDA) breaks down.
We use the Bogoliubov-de Gennes formalism to analyze the ground state phases of harmonically trapped two-species fermion mixtures with unequal masses. In the weakly attracting limit and around unitarity, we find that the superfluid order parameter is spatially modulated around the trap center, and that its global maximum occurs at a finite distance away from the trap center where the mixture is locally unpolarized. As the attraction strength increases towards the molecular limit, the spatial modulations gradually disappear while the Bardeen-Cooper-Schrieffer (BCS) type nonmodulated superfluid …
We consider trapped atomic Fermi gases with Feshbach-resonance enhanced interactions in pseudogap and superfluid temperatures. We calculate the spectrum of RF(or laser)-excitations for transitions that transfer atoms out of the superfluid state. The spectrum displays the pairing gap and also the contribution of unpaired atoms, i.e. in-gap excitations. The results support the conclusion that a superfluid, where pairing is a many-body effect, was observed in recent experiments on RF spectroscopy of the pairing gap.
We consider the problem of spin-triplet p-wave superfluid pairing with total spin projection $m_s=0$ in atomic Fermi gas across the Feshbach resonance. We allow for imbalanced populations and take into account the effects due to presence of a parabolic trapping potential. Within the mean-field approximation for the one- and two-channel pairing models we show that depending on the distance from the center of a trap at least two superfluid states will have the lowest energy. Superfluid shells which emerge in a trap may have two out of three angular components of the p-wave superfluid order param…
Interacting Fermi gases with equal populations but unequal masses are investigated at zero temperature using local density approximation and the hydrodynamic theory of superfluids in the presence of harmonic trapping. We derive the conditions of energetic stability of the superfluid configuration with respect to phase separation and the frequencies of the collective oscillations in terms of the mass ratio and the trapping frequencies of the two components. We discuss the behavior of the gas after the trapping potential of a single component is switched off and show that, near a Feshbach resona…
We consider the effect of the indirect interaction due to the exchange of density fluctuations on the critical temperature of superfluid $^{6}$Li . We obtain the strong coupling equation giving this critical temperature. This equation is solved approximately by retaining the same set of diagrams as in the paramagnon model. We show that, near the instability threshold, the attractive interaction due to density fluctuations gives rise to a strong increase in the critical temperature, providing a clear signature of the existence of fluctuation induced interactions.
Trapped ultracold Fermi gases provide a system that can be tuned between the BCS and BEC regimes by means of a magnetic-field Feshbach resonance. Condensation of fermionic atom pairs in a $^{40}$K gas was demonstrated experimentally by a sweep technique that pairwise projects fermionic atoms onto molecules. In this paper we examine previous data obtained with this technique that probed the phase boundary in the temperature-magnetic field plane. Comparison of the $^{40}$K data to a theoretically computed phase diagram demonstrates good agreement between the two.
We investigate theoretically the properties of a trapped gas of fermionic atoms in both the normal and the superfluid phases. Our analysis, which accounts for the shell structure of the normal phase spectrum, identifies two observables which are sensitive to the presence of the superfluid: the response of the gas to a modulation of the trapping frequency, and the heat capacity. Our results are discussed in the context of experiments on trapped Fermi gases.
The attractive Fermi-Hubbard Hamiltonian is solved via the Bogoliubov-de Gennes formalism to analyze the ground state phases of population imbalanced fermion mixtures in harmonically trapped two-dimensional optical lattices. In the low density limit the superfluid order parameter modulates in the radial direction towards the trap edges to accommodate the unpaired fermions that are pushed away from the trap center with a single peak in their density. However in the high density limit while the order parameter modulates in the radial direction towards the trap center for low imbalance, it also m…
We investigate the possibility that the broken spatial inversion symmetry by a trap potential induces a spin-triplet Cooper-pair amplitude in an $s$-wave superfluid Fermi gas. Being based on symmetry considerations, we clarify that this phenomenon may occur, when a spin rotation symmetry of the system is also broken. We also numerically confirm that a triplet pair amplitude is really induced under this condition, using a simple model. Our results imply that this phenomenon is already present in a trapped $s$-wave superfluid Fermi gas with spin imbalance. As an interesting application of this p…
In order to help detecting superfluidity, we theoretically investigate p-wave pairing superfluids in neutral Fermion atom gases confined by a three dimensimentional (3D) harmonic potential. The Ginzburg-Landau framework, which is generic for p-wave superfluids, is used to describe the order parameter spatial structure, or texture characterized by the l-vector both at rest and under rotation. The l-vector configuration is strongly contrained by the boundary condition due to a trap. It is found that the ground state textures exhibit spontaneous supercurrent at rest both cigar and pancake shape t…
The density distributions of the two components of a trapped, ultracold Fermi gas with population imbalance reveal the effect of imbalance on superfluid pairing. We develop a path-integral derivation of the density, that takes into account both fluctuations beyond mean-field and effects of nonzero temperature. The calculated density profiles compare favorably to the measured density profiles, and illustrate the necessity to incorporate both quantum fluctuations and finite temperature effects. The temperature dependence of the density profiles, especially near the superfluid-normal interface, a…
A trapped degenerate Bose gas exhibits superfluidity with spatially nonuniform superfluid density. We show that the vortex distribution in such a highly inhomogeneous rotating superfluid is nevertheless nearly uniform. The inhomogeneity in vortex density, which diminishes in the rapid-rotation limit, is driven by the discrete way vortices impart angular momentum to the superfluid. This effect favors highest vortex density in regions where the superfluid density is most uniform (e.g., the center of a harmonically trapped gas). A striking consequence of this is that the boson velocity deviates f…
We present a method to determine the critical rotational frequencies for superfluidity of both uniform and trapped Fermi gases across wide Feshbach resonance. It is found that as one approaches the resonance from the BCS side, beyond a critical scattering length, pairing is so robust that superfluidity cannot be destroyed by rotation. Moreover, the critical frequency has a sequence of jumps revealing the appearance of Landau levels, which are particularly prominent for systems up to a few thousand particles. For rotational frequency below an "ultimate" critical frequency, defined to be the low…
Starting with the two-band description of an orbital Feshbach resonance, we study superfluid properties of a trapped $^{173}$Yb Fermi gas under the assumptions of a local-density approximation for the trapping potential and a mean-field approximation for the intra-band Cooper pairings. In particular, we investigate the competition and interplay between the pair-breaking effect that is caused by the inter-band detuning energy, and the pair-breaking and thermal-broadening effects that are simultaneously caused by the temperature. We predict several experimental signatures that are directly cause…
We theoretically investigate the ground state of trapped neutral fermions with population imbalance in the BCS-BEC crossover regime. On the basis of the single-channel Hamiltonian, we perform full numerical calculations of the Bogoliubov-de Gennes equation coupled with the regularized gap and number equations. The zero-temperature phase diagram in the crossover regime is presented, where the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing state governs the weak-coupling BCS region of a resonance. It is found that the FFLO oscillation vanishes in the BEC side, in which the system under populati…
The physical properties of arbitrary half-integer spins $F = N - 1/2$ fermionic cold atoms trapped in a one-dimensional optical lattice are investigated by means of a low-energy approach. Two different superfluid phases are found for $F \ge 3/2$ depending on whether a discrete symmetry is spontaneously broken or not: an unconfined BCS pairing phase and a confined molecular superfluid instability made of $2N$ fermions. We propose an experimental distinction between these phases for a gas trapped in an annular geometry. The confined-unconfined transition is shown to belong to the $Z_N$ generaliz…
We investigate the superfluid-to-normal zero temperature quantum phase transitions of asymmetric two-component Fermi gases as a function of the chemical potential imbalance $h$. The calculations are performed for homogeneous and trapped imbalanced systems. We concentrate at unitarity, characterized by a divergent interaction parameter $k_F a$, where most of the current experiments are realized. For homogeneous systems, we determine the critical chemical potential imbalance $h_c$ at which possible phase transitions occur. In the case of trapped gases, we show how $h_c$ can be consistently deter…
We study the superfluid transition in a quasi-two-dimensional Fermi gas with a magnetic field tuning through a Feshbach resonance. Using an effective two-dimensional Hamiltonian with renormalized interaction between atoms and dressed molecules, we investigate the Berezinskii-Kosterlitz-Thouless transition temperature by studying the phase fluctuation effect. We also take into account the trapping potential in the radial plane, and discuss the number and superfluid density distributions. These results can be compared to experimental outcomes for gases prepared in one-dimensional optical lattice…
In this chapter we review recent experimental and theoretical work on various novel superfluid phases in fermion systems, that result from pairing fermions of different species with unequal densities. After briefly reviewing existing experimental work in superconductors subject to a strong magnetic field and trapped cold fermionic atom systems, we discuss how to characterize the possible pairing phases based on their symmetry properties, and the structure/topology of the Fermi surface(s) formed by the unpaired fermions due to the density imbalance. We also discuss possible experimental probes …
We discuss an experimental technique to visualize the motion in the bulk of superfluid $^4$He by tracking micron-sized solid hydrogen tracers. The behavior of the tracers is complex since they may be trapped by the quantized vortices while also interacting with the normal fluid via Stokes drag. We discuss the mechanism by which tracers may be trapped by quantized vortices as well as the dependencies on hydrogen volume fraction, temperature, and flow properties. We apply this technique to study the dynamics of a thermal counterflow. Our observations serve as a direct confirmation of the two-flu…
We study the finite temperature $T$ behavior of trapped Fermi gases as they undergo BCS-Bose Einstein condensation (BEC) crossover, in the presence of a population imbalance. Our results, in qualitative agreement with recent experiments, show how the superfluid phase transition is directly reflected in the particle density profiles. We demonstrate that at $ T \neq 0$ and in the near-BEC and unitary regimes, the polarization is excluded from the superfluid core. Nevertheless a substantial polarization fraction is carried by a normal region of the trap having strong pair correlations, which we a…
We investigate strong coupling effects on the superfluid phase transition in a gas of Fermi atoms with a Feshbach resonance. The Feshbach resonance describes a composite quasi-Boson, which can give rise to an additional pairing interaction between the Fermi atoms. This attractive interaction becomes stronger as the threshold energy of the Feshbach resonance two-particle bound state is lowered. In a recent paper, we showed that in the uniform Fermi gas, this tunable pairing interaction naturally leads to a BCS-BEC crossover of the Nozi`eres and Schmitt-Rink kind, in which the BCS-type superflui…
Vortex arrays in type-II superconductors admit the translational symmetry of an infinite system. There are cases, however, like ultra-cold trapped Fermi gases and the crust of neutron stars, where finite-size effects make it quite more complex to account for the geometrical arrangement of vortices. Here, we self-consistently generate these arrays of vortices at zero and finite temperature through a microscopic description of the non-homogeneous superfluid based on a differential equation for the local order parameter, obtained by coarse graining the Bogoliubov-de Gennes (BdG) equations. In thi…
We explore theoretically the novel superfluidity of harmonically-trapped polarized ultracold fermionic atoms in a two-dimensional (2D) optical lattice by solving the Bogoliubov-de Gennes equations. The pairing amplitude is found to oscillate along the radial direction at low particle density and along the angular direction at high density. The former is consistent with the existing experiments and the latter is a newly predicted Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, which can be tested in experiments.
We study the discrete nonlinear Schrödinger equation (DNLS) in an annular geometry with on-site defects. The dynamics of a traveling plane-wave maps onto an effective ''non-rigid pendulum'' Hamiltonian. The different regimes include the complete reflection and refocusing of the initial wave, solitonic structures, and a superfluid state. In the superfluid regime, which occurs above a critical value of nonlinearity, a plane-wave travels coherently through the randomly distributed defects. This superfluidity criterion for the DNLS is analogous to (yet very different from) the Landau superfluidity…
